In a compatible finite element model, variables in the discretisation are chosen to lie in the spaces of the discrete de Rham complex that correspond to their continuous analogues.
In this structure, the discrete differential operators preserve vector calculus identities such as \(\mathbf {\nabla }\times \mathbf {\nabla }f=\mathbf {0}\)  for all scalar \(f\) .
Applied to fluid dynamics, this structure suggests that the velocity should lie in a \(H(\mathrm {div})\) -conforming space such as the \(\mathrm {RTc}^f_k\)  elements.
As mentioned in Section , it was shown by [1]} that the choice of \(\mathrm {RTc}^f_k\) -\(\mathrm {DG}_{k-1}\)  for the wind and height fields on quadrilateral elements in a shallow-water model gives a discretisation with many desirable properties.
As explained by [1]}, this pair of elements has the optimal ratio of degrees of freedom (DoFs) for capturing the wave modes of the shallow-water equations, and it mimics the properties of the popular C-grid staggering used in finite difference models.
For this reason, a similar discretisation will be used in the Met Office's new LFRic model, with the wind lying in the three-dimensional form of \(\mathrm {RTc}^f_1\)  space.
Although the use of the corresponding Raviart-Thomas elements on triangular elements in a shallow-water model has been investigated elsewhere, notably by [3]} as part of the \(\mathrm {RT}^f_k\) -\(\mathrm {DG}_{k-1}\)  pair, as shown by [1]} it suffers from inferior representation of the shallow-water wave modes.
Given this result, this work focuses on quadrilateral elements.
